EPILOGUE-LIKE STUFF: explaining this alternate universe…
Okay, first off, both Charlie and Bess do go to Hogwarts. All three Dursley children eventually marry other magical persons and settle in the magical community. While at school Billy wound in up Ravenclaw, Bess in Hufflepuff, and – to Harry's amusement, only Charles wound up in Gryffindor (though, as in Harry's case, the hat apparently considered Slytherin, but only briefly.) The Dursley children are very friendly with their Potter cousins, but had close friends of their own while at Hogwarts.
In my alternate universe, you didn't see the Dursleys in the Epilogue of "Deathly Hallows" because, the year he sent his second son, Albus, to Hogwarts for the first time, Harry was running late and didn't have much time to deal with anything but his children's concerns before the train left. After the express chugged out, Harry and Dudley spotted each other and commiserated, as Dudley also had a youngest child pouting at being left behind (Bess is a "year and a quarter" younger than Charles, so Dudley's pain was shorter-lived than Harry's)
But the bigger questions you might have are "If the Dementors didn't give Dudley a personality transplant, who did?" and "How did Petunia grow a backbone that late in life?" The answers are related.
As we saw at the end of "Deathly Hallows," Dudley experienced a change of heart toward Harry the summer before Harry's 5th year at Hogwarts when Harry saved him from the Dementors and carried him home. His changed view of Harry and Harry's relative worth as a human being (let alone the relative worth of the "abnormality" of magic) was the first domino to fall in a series of often uncomfortable personal revelations.
If Harry wasn't a "waste of space," then was it right to treat him the way his family had? If Harry wasn't a "waste of space" who else wasn't? Was it right to treat anyone that way? Even if magic was "abnormal" why would his mother (for all he knew or understood) have shut-out her own sister – her only remaining living relative aside from Harry?
Dudley reached no firm conclusions, though these uncomfortable thoughts were leading to small changes in his behavior. Then, in spite of how hatefully the Dursleys continued to treat Harry, he still worked hard to persuade them to flee to safety. Dudley was impressed with Harry, in spite of himself. On the other hand, Vernon's determination to remain blind, in spite of the very real threat to his family's lives, gave Dudley further distressing insight into the "Dursley World View" he was coming to question. These uncomfortable insights were strengthened by how poorly his father treated the witch and wizard who were sent to help and protect them.
Uncomfortable with his altered view of his parents, particularly his father, Dudley tried to limit the time he spent with them during the time they were in hiding. Instead, he got to know the witch and wizard who took turns checking in on them daily. He also learned a lot from the Squib who came to tutor him so he could take his A-Levels.
Vernon remained as determined as ever to see the world the way he wanted to see it. He, therefore, didn't really notice much of a change in Dudley. After all, he was used to his son NOT being around most of the year while he was away at school, and Dudley had never spent much time interacting with his parents while on school holiday. So, as long as Dudley ate most evening meals with his parents, Vernon remained happily oblivious.
Petunia, however, was a different story. Her commitment to the Dursley World View was stronger and longer-standing than Dudley's. It was also fueled by bitter disappointment and fear. Still, the last several years had shown there were a few fault lines in it. If she let herself dwell on things like the Dementor incident she had to admit Harry wasn't all that horrible… so she refused to think about it.
She did notice changes in Dudley, the most pertinent of which was his clear preference to distance himself from Vernon. Petunia was forced to admit (because it affected her, after all) her fear that if Dudley wasn't forced to remain in their safe house with them, he seemed would take off and get as far away as possible.
Here's the thing about Petunia: she invested herself and her life into the Dursley World View, becoming exactly what Vernon wanted without complaint. (Ah, the foolish "I'll show you" choice of the most non-magical person she could find!) Accepting this to be the life she had chosen, she committed herself to it without complaint or second thought. It even seemed a rather wise choice when her sister and sister's husband were murdered by an evil wizard. She would have stayed unquestioningly in that course, in spite of the difficulties "raising" Harry presented.
But now the one born into the Dursley World View was questioning it with a rather serious intensity. He wanted as little to do with his parents as possible and his leaving seemed to only be a matter of time. When they left hiding, would Dudley leave them for good? So, Petunia was faced with a dilemma. What was more important to her: staying the course with the Dursley World View or keeping her son in her life? Petunia may be a woman of many faults, but indifference to her only child is not one of them! As Harry correctly guessed in the story, if she had to chose between her husband and her son, Vernon would lose by a mile.
Harry's defeat of Voldemort released the Dursleys from hiding in time for Dudley to take his A-levels and he did well - better than he would have if he'd returned to Smeltings for his final year. He was accepted at university and left early to find an off-campus flat and a part-time job so he had a good excuse to remain away from home during term breaks.
So, Dudley arrived at University a slightly better, if still somewhat socially inept person. He had a long way to go in terms of joining the human race, but he was on the right path. He was pretty sure he knew who he didn't want to be, but uncertain how to get to who he did want to be. In addition, in spite of the questions of the last three years, he had many years hard training to be a git… and that's hard to shake. Fortunately he wound up associating with some exceptionally decent people who helped him along with his emotional re-education.
As you might have guessed, Sarah Collins was one of those people. She saw the essential decency passed along (if only in the genetic code) from the Evans side of his family tree and gave him a chance.
Vernon steadfastly ignored the changes in his son, but Petunia wisely made room for them, while careful not to disturb Vernon's point of view (bless her heart, that would have been a job, wouldn't it?). It was Petunia who smoothed things over when Sarah became pregnant during her and Dudley's final term at university, wisely encouraging Vernon to ignore the fact that pregnancy was why Dudley got married right after graduation.
The chasm between Dudley and his father grew as Dudley did, but Vernon steadfastly ignored it. Sarah and Dudley learned early on the best way to manage Vernon was to let him think what he wanted and to keep his visits short. Though she remained with Vernon and essentially committed to the Dursley World View (a commitment made less odious by the absence of Harry), Petunia's most important and truest allegiance was cemented with the birth of her first grandchild. When Petunia held William in her arms she knew nothing in the world would ever be more important to her.
Spending time over the years at her Son and daughter-in-laws home without Vernon (who, thankfully, had work commitments) allowed her to even show a few signs of the person she might have become with different choices. Don't get me wrong, she could still be quite revolting as a grandmother (see the opening lines of chapter one) and annoying as a mother-in-law. Still, her devotion to her grandchildren allowed her to re-discover her dormant decency.
She's still a bit too much beholden to appearances so she'll stick with Vernon if he doesn't force her to choose… fortunately she's learned to mange him.
And Dudley? He occasionally shows shades of the person he no longer wants to be (don't we all?) but he turned out to be a pretty decent chap… and an excellent father. After all, in case you hadn't noticed, Sarah wouldn't have it any other way. As Harry so artfully put it "she'd rip his guts out with her bare hands" if he wasn't… and who wants that?